clt

Central Limit Theorem

CLT states that the distribution of ${\bar{X}}_n$ after standardization approaches a standard Normal distribution.

Central Limit Theorem

As $n\to \infty$,

$$\sqrt{n}\left(\frac{{\bar{X}}_n-\mu }{\sigma }\right)\to \mathcal{N}(0,1)$$

in distribution.

Regardless the distribution of $X_j$, averaging these r.v.s will always lead to normality.

Consider the sum of the i.i.d. r.v.s, we also have

$$W_n\sim \mathcal{N}(n\mu ,n{\sigma }^2)$$

Which is useful for approximating sum of i.i.d. r.v.s.

• Poisson: $Y\sim \mathcal{N}(n,n)$
• Binomial: $Y\sim \mathcal{N}(np,np(1-p))$

Cauchy distribution has no true mean and variance, and CLT and LLN doesn’t hold.